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CHAPTER VII.

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Quantities of Materials (Cont'd).

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II. Volumes from Original Contours.

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2. Grading Over Extended Areas.

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Application of the Prismoidal Formula.

The general formula for the value of any number of continuous prismoids
has
been shown to be
V cu. ft. = (2 × L) ÷ 3 (A_{0} + 2 A_{1}
+ A_{2} + 2 A_{3} + A_{4} + ... 2 A_{n–1}
+ A_{n} – (A_{0} + A_{n}) ÷ 2) ... (19)

or
V cu. yds. = (2 × L) ÷ (3 × 27) (A_{0}
+ 2 A_{1} + A_{2} + 2 A_{3} + A_{4} + ...
2 A_{n–1} + A_{n} – (A_{0} + A_{n}) ÷
2) ... (20)
In the particular problem we are using for
illustration there are 10 small constituent prismoids each 2 ft. long.

In applying Eq. 20 to this particular case it is evident that the areas
ABC and ARSTV of Fig. 1 Plate X are the A_{0}
and A_{n} of that Equation and since there are 10 of these small
prismoids each 2 ft. long *n* will be 10 and L = 2.0.

Designating the areas of the bases of these 10 constituent prismoids
by the numbers of the contours on the diagram
we have the areas 18, 14, 10, 6, 2 as the values of the quantities A_{1},
A_{3}, A_{5}, A_{7} and A_{9}, respectively
or the odd A's of Eq. 20, while the areas 20, 16, 12, 8, 4 and 0 are the
corresponding values of A_{0}, A_{2}, A_{4}, A_{6},
A_{8} and A_{10} or the *even* A's of that Equation.

Substituting all these values in Eq. 20 we have as the total volume
of excavation

V = (2 × 2) ÷ (3 × 27) ((20) + 2(18) + (16)
+ 2(14) + (12) + 2(10) + (8) + 2(6) + (4) + 2(2) + (0) – ((0) + (20)) ÷
2) ... (a)
or generally
V = (2 × 2) ÷ (3 × 27) (Sum Even A's + 2 Sum
Odd A's – Sum Extreme A's ÷ 2) ... (b)
Since our diagram is plotted to a scale of
1" = 10', 1 Sq.. Inch = 10² = 100 sq. Ft., which being introduced
into Eq. (a) gives for the total volume after reduction.

V = 4.9382 (Sum Even A's + 2 Sum Odd A's – Sum Extreme A's ÷
2) ... (c)
By assuming the expression within the parenthesis of Eq. (c) to be 1
Sq. inch of actual area we have

V = 4.9382 Cu. Yds. ... (d)
Eq. (d) shows that after having performed all of the operations indicated
by the expression within the parenthesis of Eq. (a) each square inch of
actual area indicated by the result of those operations when plotted to
a scale of 10 ft. to 1 inch will represent a volume of 4.9382 Cu. Yds.